A laptop can turn off for various reasons. Components could be broken or overheated; the circuits could be over- or under-powered or even short circuited. There are some protection mechanisms in place that immediately turn off the laptop when something out of the specifications is measured. You need to diagnose the circuits that supply power, consume them and produce heat.
First clear out the power to make sure the protection mechanisms are reset, as described in your question. Then, you measure the voltage and ampere of the power supply. In a sufficiently disassembled laptop it is also possible to further measure the power circuit near the power connector.
To diagnose whether or not it is the motherboard or something attached to it you need to start unplugging anything that is removable such that you get an idea as to whether that allows the system to boot further or not. Also check if there is too much heat in the cooling system near the CPU/GPU/Transistors/Caps/... in case your system has turned off at a non-POST time.
Finally, when you put them back ensure that the contacts are clean and all screws are back in place.
1Most likely the power supply unit is dead and needs replacement. – harrymc – 2015-11-05T20:44:36.953
Can you turn on the laptop if it is connected to a socket? (Loading Cable) – RookieTEC9 – 2015-11-05T21:07:48.193
@RookieTEC9 No, it does not turn on when connected to a socket. – Joshua Fox – 2015-11-06T07:27:03.677
@harrymc How do I diagnose that the power supply is dead as opposed to some other cause (like a disconnection somewhere in the internal power network.) – Joshua Fox – 2015-11-06T07:28:02.590
A professional can measure the electric current on entry and exit of the PSU. But the simplest test is to try another PSU. If you know how to open the laptop (I suppose it's no longer under warranty), you could even use a desktop PSU as long as as it has at least as much capacity - the PSU doesn't have to go into the case, only the wires do (this won't work if the PSU is soldered to the motherboard). Otherwise, get help from a knowledgeable friend or a professional repairman. – harrymc – 2015-11-06T08:11:23.547